Suppose \(L\)  is an oriented link with \(n\)  components.
Take a planar diagram for \(L\) . Numerating arbitrarily, let \(x_1,\ldots ,x_m\)  be the directed arcs, and \(\kappa _1,\ldots ,\kappa _m\)  the crossings.
Let \(F_m\)  be the free group on the letters \(\tilde{x}_1,\ldots ,\tilde{x}_m\) , with \(\tilde{x}_j\)  corresponding to \(x_j\) .
With the “over presentation" (see [1]} Chapter VI) for \(\pi =\pi _1(L)\) ,
each positive crossing as in Figure REF  (a) gives rise to a relator \(\tilde{x}_j\tilde{x}_k\tilde{x}_j^{-1}\tilde{x}_\ell ^{-1}\) , and each negative crossing as in Figure REF  (b) gives rise to a relator \(\tilde{x}_j^{-1}\tilde{x}_k\tilde{x}_j\tilde{x}_\ell ^{-1}\) . By abuse of notation, we always denote the element of \(\pi \)  represented by a directed arc \(x\)  also by \(x\) . Let \(\phi :F_m\twoheadrightarrow \pi \)  be the quotient map sending \(\tilde{x}_j\)  to \(x_j\) . Let \(\widetilde{M}\in \mathcal {M}_m(\mathbb {Z}F_m)\)  be the matrix whose \((i,j)\) -entry is \(\partial \tilde{r}_i/\partial \tilde{x}_j\)  (the Fox derivative),
and let \(M=\check{\phi }(\widetilde{M})\in \mathcal {M}_{m}(\mathbb {Z}\pi )\) .
Let
\(\alpha :\pi \twoheadrightarrow H_1(S^3-L)\cong \mathbb {Z}^n=\langle t_1,\ldots ,t_n\rangle \) 
